It’s easier to get your head around this scenario if we start with a much simpler version of this, moving at much slower speeds. So let’s say we have a convertible, driving at 40 miles an hour, and a passenger in that car can throw a ball at 10 miles an hour. If the passenger throws the ball straight up while the car is moving, the passenger can catch that ball when it comes back down. Someone observing this scene from the side of the road would say that the ball is moving to the side along with the car, while the passenger inside the car would tell you that the ball didn’t move horizontally relative to the car. (This means that when the ball came back down, it landed in the hands of the person throwing it, instead of hitting the front or back of the car.) Both statements are perfectly correct, and both the side of the road watcher and the passenger in the car would tell you that the ball flew upwards at 10 miles an hour, and then back down.
This kind of thought experiment illustrates an important point in physics - motion in the horizontal direction and motion in the vertical direction can be treated completely separately from each other. If I’m just interested in describing the motion of the ball up and down, that can be done regardless of what the horizontal motion of that ball is doing - any horizontal motion simply takes a vertical path up and down and stretches it out sideways.
Once we speed things up to fractions of the speed of light, the concepts of special relativity begin to apply, but this basic division of motion remains constant. Two things do change, though, and the first is that we need to be much more careful with where our watchers are when we describe what it is that they see. The other change is that the conversion from what the passenger in the convertible sees to what the person on the side of the road sees is more complex than simply remembering to add in the motion of the car.
If our shuttle exits the larger spaceship, and has two rockets on it, one on each end, so that it can move at a speed of one-tenth the speed of light at a perfect 90 degrees, and then stop, and go in reverse back to the spaceship, we have effectively replicated our slower situation. A person on the spaceship would say that the shuttle isn’t moving at all horizontally (in the direction the spaceship is traveling) but is bouncing outwards and back without shifting along the spaceship from its berth. Because it’s moving at 90 degrees to the direction the spaceship is traveling, there isn’t anything that would prevent the shuttle from coming straight back to its dock on the main spaceship.
An observer on a nearby planet would see the spaceship passing by at four tenths the speed of light, and they would see a shuttle leave the main craft, appear to drift outward at an angle, and then drift back inwards, meeting back up with the main craft. That planet based observer would measure the speed of the shuttle to be higher than 0.1c, because it’s got a horizontal speed that the observer on the spaceship doesn’t see, along with its motion at a right angle to the spaceship.
But what if the shuttle doesn’t go out at a right angle to the spaceship’s direction of travel? In that case, what the watching people see on the spaceship and on the planet they’re zipping past might differ a little more. If our shuttle can reverse directions instantaneously, going from 0.1c “forwards” with the spaceship to 0.1c “backward” right away, then someone on the spaceship would simply see the shuttle moving at 0.1c relative to the ship, which is stationary under their feet. As long as the shuttle is always moving at a fixed speed, it should be able to reverse course and catch back up to its dock.
The planetary observer, meanwhile, would observe a whole host of things changing. (And if we wanted to make things super complicated, we could look at how much time each set of observers is dealing with.) Velocities don’t add in the same way when you’re moving at significant fractions of the speed of light, and so the shuttle would appear to be moving at different speeds relative to the planet, depending on whether the shuttle was moving against the flow of the spaceship (0.29c) or along with it (0.48c).
The shuttle could wind up in a situation where it couldn’t reach the spaceship again if it could slow itself down enough. In the above scenario, I’ve assumed that that shuttle is always moving at a fixed speed, but if the shuttle could change its speed so that instead of being stationary relative to the spaceship, it were stationary relative to the planet, it would not be able to accelerate back up to the speed of the spaceship. Then it would be stuck, lagging ever further behind the spaceship it came from.
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